On $k$-graceful labeling of pendant edge extension of complete bipartite graphs

For any two positive integers $m,n$, we denote the graph $K_{m,n}\odot K_1$ by $G$. Ma Ke-Jie proposed a conjecture [9] that pendant edge extension of a complete bipartite graph is a $k$-graceful graph for $k \ge 2$. In this paper we prove his conjecture for $n\le m < n^2+\lfloor\frac{k}{n}\rfloor+ r$.